Figure 5. A schematic representation of a function fixlt x2) subject to constraint g(xlt x2) = 0, which restricts solutions to the shaded area only.

Figure 5. A schematic representation of a function fixlt x2) subject to constraint g(xlt x2) = 0, which restricts solutions to the shaded area only.

solved in combination with g(xly x2) = 0 to locate the stationary point (5).

The preceding simple two-dimensional case generalizes to the case of n independent variables subject to m < n constraints. Applying the same reasoning as before produces the so-called Jacobian determinants, which in turn can be used with the constraint equation 9 to obtain the stationary points (5).

The most commonly used of the three analytical methods mentioned is Lagrange multipliers. Its simplest form is the two-dimensional case. Multiplying equation 12 by a constant X and adding the result to equation 11 yields

Equation 13 transforms to

CrXi OX 2

The constant X is the Lagrange multiplier of the system and is determined by equating the bracketed terms of equation 13 to zero. This produces (see equation 14) the augmented function A

which guarantees that the stationary point(s) of the system will be located if X is considered to be an independent variable and the following system of equations is solved.

In the general case of n independent variables with m constraint equations, the augmented function takes the form f + Xig! + . . . + X„gm

and a system of (n + m) equations must be solved to locate the stationary points. This is usually easier than a solution by direct substitution or constraint variation, unless the system is very simple.

Optimization of Systems with Inequality Constraints. Scientists and engineers understand that a profound difference exists between mathematical abstraction and physical reality. In the world of mathematics, an equality sign represents perfect equality, but in the real world equality requirements are physically impossible and unrealistic. For example, constraints such as net weight = 425 g or room temperature = 20°C are difficult to realize. However, targets such as contain no less than 425 g, or room temperature less than 20°C, are easier to achieve. Thus inequalities are an inherent part of the real world and re quire attention. With respect to optimization, several techniques exist to solve problems with inequality constraints.

If the system under consideration is linear, linear programming can be used to optimize it. The term programming here does not refer to computer programming, but rather to mathematical programming and scheduling. Several explanations of the mathematical relations involved in linear programming are available (12,16-21).

In some cases, it might be interesting to find out how a solution to a linear programming problem changes as the problem's data change in systematic and predetermined ways. Parametric programming can solve this problem. The algorithm for obtaining solutions of parametric programming problems has been published (12). In other cases, the objective function might not be linear, but quadratic instead. In such cases, quadratic programming can determine the optima (16). In more general terms, either the objective function or some (or all) of the constraints or both might be nonlinear in nature. Methods of nonlinear programming must then be applied (12,18). Other types of mathematical programming include (12,16) integer programming, which deals with optimization problems whose variables (some or all) are required to take integer values (ie, the number of machines or components required for some processing system). There is also geometric programming, which solves nonlinear programming problems containing special functions constructed from terms of the form

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